A resolvent approach to traces and zeta Laurent expansions

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Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TR A introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C₀(A,P) in the trace expansion of A(P-λ)⁻ⁿ (with an auxiliary elliptic operator P), as determined in a joint work with Seeley 1995. The definition of TR A is extended from the cases of noninteger order, or integer order and even-even parity on odd-dimensional manifolds, to the case of even-odd parity on even-dimensional manifolds. For the generalized zeta function ζ(A,P,s)=(AP⁻ˢ), extended meromorphically to C, C₀(A,P) equals the coefficient of s⁰ in the Laurent expansion at s=0 when P is invertible. In the mentioned parity cases, ζ(A,P,s) is regular at all integer points. The higher Laurent coefficients Cⱼ(A,P) at s=0 are described as leading nonlocal coeficients C₀(B,P) in trace expansions of resolvent expressions B(P-λ)⁻ⁿ, with B log-polyhomogeneous as defined by Lesch (here -C₁(I,P)=C₀(P,P) gives the zeta-determinant). C₀(B,P) is shown to be a quasi-trace in general, a canonical trace TR B in restricted cases, and the formula of Lesch for TR B in terms of a finite part integral of the symbol is extended to the parity cases.